Existential quantifier. This is denoted by ∃ and is read as “there exists a...”
∃I ∈ {7,8,9}IsPrime(I)
This quantified expression can be read as:
there exists a value in the set {7,8,9} which satisfies the truth-valued function IsPrime.
The expression
∃I ∈ {11,12,13}IsOdd(I)
is indeed true as it is equivalent to the proposition:
IsOdd(11) ∨IsOdd(12) ∨IsOdd(13)
The reason that quantifiers extend the expressive power of the logic is that the sets in the constraint of a quantified expression can be infinite. Such an expression abbreviate a disjunction which could never be completely written. For example:
∃I ∈ ℕ1 IsPrime(I)
or:
∃I ∈ ℕ1 ¬IsPrime(2I − 1)
express facts about prime numbers.
Just as a disjunction can be viewed as an existentially quantified expression, a conjunction such as:
IsEven(2) ∧IsEven(4) ∧IsEven(8)
can be written as a universally quantified expression:
∀I ∈{2,4,8}IsEven(I)
here again, universal quantification increase the expressive power of the logic when we deal with infinite sets:
∀I ∈ ℕ IsEven(2 ∗ I)
We can now formally define IsPrime as
IsPrime(I) ≜ | I≠1 ∧ |
∀D ∈ ℕ1 D divides I ⊃ D = 1 ∨ D = I |